Problem 42 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrr}0.3 & 1.1 & 2.4 \\\0.9 & -0.1 & 0.4 \\\\-0.7 & 0.3 & -0.5\end{array}\right] \quad B=\left[\begin{array}{cc}1.2 & -0.1 \\\0 & -0.5 \\\0.5 & -2.1\end{array}\right]$$ $$C=\left[\begin{array}{rrr}-0.2 & 0.2 & 0.1 \\\1.1 & 2.1 & -2.1\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operation, or explain why it cannot be performed. $$A^{2}$$

Step-by-Step Solution

Answer

The square of matrix A is \(\begin{bmatrix}-0.60 & 0.94 & -0.04 \\ -0.10 & 1.12 & 1.92 \\ 0.41 & -0.95 & -1.31\end{bmatrix}\).

Step 1: Determine if A squared is possible

To find the square of a matrix, the operation involves multiplying the matrix by itself. Let's first check if the matrix multiplication is possible. For the multiplication of two matrices to be feasible, the number of columns in the first matrix must match the number of rows in the second matrix. Matrix \(A\) is a \(3 \times 3\) matrix, meaning it has 3 columns and 3 rows, so the multiplication \(A \cdot A\) can be performed.

Step 2: Matrix Multiplication Setup

To compute \(A^2 = A \cdot A\), write down matrix \(A\) twice:\[A = \begin{bmatrix}0.3 & 1.1 & 2.4 \0.9 & -0.1 & 0.4 \ -0.7 & 0.3 & -0.5\end{bmatrix}\]Set up the multiplication to find the resulting \(3 \times 3\) matrix. Each element \(c_{ij}\) in the product matrix \(C\) is calculated by taking the dot product of the \(i\)-th row of the first \(A\) with the \(j\)-th column of the second \(A\).

Step 3: Calculate Elements of the First Row of Result Matrix

- For element \(c_{11}\), compute: \((0.3 \times 0.3) + (1.1 \times 0.9) + (2.4 \times -0.7) = 0.09 + 0.99 - 1.68 = -0.60\).- For element \(c_{12}\), compute: \((0.3 \times 1.1) + (1.1 \times -0.1) + (2.4 \times 0.3) = 0.33 - 0.11 + 0.72 = 0.94\).- For element \(c_{13}\), compute: \((0.3 \times 2.4) + (1.1 \times 0.4) + (2.4 \times -0.5) = 0.72 + 0.44 - 1.20 = -0.04\).

Step 4: Calculate Elements of the Second Row of Result Matrix

- For element \(c_{21}\), compute: \((0.9 \times 0.3) + (-0.1 \times 0.9) + (0.4 \times -0.7) = 0.27 - 0.09 - 0.28 = -0.10\).- For element \(c_{22}\), compute: \((0.9 \times 1.1) + (-0.1 \times -0.1) + (0.4 \times 0.3) = 0.99 + 0.01 + 0.12 = 1.12\).- For element \(c_{23}\), compute: \((0.9 \times 2.4) + (-0.1 \times 0.4) + (0.4 \times -0.5) = 2.16 - 0.04 - 0.20 = 1.92\).

Step 5: Calculate Elements of the Third Row of Result Matrix

- For element \(c_{31}\), compute: \((-0.7 \times 0.3) + (0.3 \times 0.9) + (-0.5 \times -0.7) = -0.21 + 0.27 + 0.35 = 0.41\).- For element \(c_{32}\), compute: \((-0.7 \times 1.1) + (0.3 \times -0.1) + (-0.5 \times 0.3) = -0.77 - 0.03 - 0.15 = -0.95\).- For element \(c_{33}\), compute: \((-0.7 \times 2.4) + (0.3 \times 0.4) + (-0.5 \times -0.5) = -1.68 + 0.12 + 0.25 = -1.31\).

Step 6: Write Out the Resulting Matrix

The resulting matrix \(A^2\) from multiplying \(A\) by itself is:\[A^2 = \begin{bmatrix}-0.60 & 0.94 & -0.04 \ -0.10 & 1.12 & 1.92 \ 0.41 & -0.95 & -1.31\end{bmatrix}\]

Key Concepts

Square of a Matrix

When we talk about the square of a matrix, we're referring to the operation where a matrix is multiplied by itself. This is different from squaring numbers because matrices require special rules for multiplication.
For this to happen, the matrix must be a square matrix, which means it has the same number of rows as columns. For example, our matrix \(A\) is a \(3 \times 3\) matrix. This allows us to calculate \(A^2\).
Multiplying a matrix by itself involves performing multiple calculations to obtain new elements in a resulting matrix. Each new element is the dot product of a row from the first matrix and a column from the second matrix (which is the same matrix here because we're squaring). This operation is common especially in applications that involve repeated transformations, like in computer graphics.

Dot Product

The dot product is an essential component of matrix multiplication. It's about combining vectors to form a single number. Here's how it works, especially in the context of matrices:

• Take a row from the first matrix.
• Take a corresponding column from the second matrix.
• Multiply the entries in pairs and sum them up.

Let's consider the matrix \(A\) again. If you want to find the first element of \(A^2\), denoted as \(c_{11}\), calculate the dot product of the first row with the first column.
Here's a little formula to keep in mind for two vectors \( \textbf{u} \) and \( \textbf{v} \): \( \textbf{u} \cdot \textbf{v} = u_1v_1 + u_2v_2 + \ldots + u_nv_n \). This is precisely what happens with the rows and columns during matrix multiplication, ensuring every entry is evaluated via the dot product.

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrix operations such as multiplication. The dimension of a matrix is described by the count of rows and columns it contains.
For instance, matrix \(A\) in our example is a \(3 \times 3\) matrix, indicating it has 3 rows and 3 columns. These dimensions dictate which operations can be performed.
When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. This rule allows the multiplication process to complete correctly, and when squaring a matrix, this requirement is inherently satisfied.

Precalculus

Precalculus serves as a foundational block for higher-level math courses, integrating concepts like algebra and geometry. When engaging with matrix operations in a precalculus context, students encounter various mathematical principles, such as those used in solving systems of linear equations or understanding geometric transformations.
Matrix multiplication itself can appear complex at first, but with practice, it becomes a familiar tool for students. It relies on the interaction of rows and columns, providing a stepping stone towards calculus, where matrices can describe more dynamic systems.

• It's about making connections between various mathematical tools.
• Understanding how transformations with matrices work lays groundwork for further studies.

As a part of the precalculus curriculum, matrix operations like squaring prepare students for the rigorous applications they might face later in mathematics or related fields.